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Indian National Mathematical Olympiad is organized by Homi Bhabha Centre for Science Education. This post is dedicated for INMO 2019 Discussion. You can post your ideas here.

Problem 1. Let $ABC$ be a triangle with $\angle BAC > 90^\circ$. Let $D$ be a point on the line segment $BC$ and $E$ be a point on the line $AD$ such that $AB$ is tangent to the circumcircle of triangle $ACD$ at $A$ and $BE$ perpendicular to $AD$. Given that $CA=CD$ and $AE=CE$, determine $\angle BCA$ in degrees

Problem 2. Let ${A_1}{B_1}{C_1}{D_1}{E_1}$ be a regular pentagon. For $2\leq n \leq 11$, let ${A_n}{B_n}{C_n}{D_n}{E_n}$ whose vertices are the midpoints of the sides of ${A_{n-1}}{B_{n-1} }{C_{n-1} }{D_{n-1}}{E_{n-1}}$. All the $5$ vertices of each of the $11$ pentagons are arbitrarily coloured red or blue. Prove that four points among these $55$ have the same colour and form the vertices of a cyclic quadrilateral.

Problem 3. Let $m$, $n$ be distinct positive integers. Prove that $\gcd (m,n) + \gcd (m+1,n+1) + \gcd (m+2,n+2) \leq 2|m-n|+1$. Further, determine when equality holds.

Problem 4. Let $n$ and $M$ be positive integers such that $M>n^{n-1}$. Prove that there are $n$ distinct primes $p_1$,,,….,$p_n$ such that $p_j$ divides $M+j$ for $1 \leq j \leq n$.

Problem 5. Let $AB$ be a diameter of a circle $\Gamma$ and let $C$ be a point on $\Gamma$ different from $A$ and $B$. let $D$ be the foot of perpendicular from $C$ on to $AB$. let $K$ be a point of the segment $CD$ such that $AC$ is equal to the semiperimeter of the triangle $ADK$. Show that the excircle of triangle $ADK$ opposite $A$ is tangent to $\Gamma$.

Problem 6. Let $f$ be a function defined from the set $\{(x,y) : x,y$ are real, $xy \neq 0\}$ to the set of all positive real number such that

(i)$f(xy,z)=f(x,z)f(y,z),$ for all $x,y \neq 0$

(ii) $f(x,1-x)=1,$ for all $x \neq 0,1$

Prove that

(i)$f(x,x)=f(x,-x)=1,$ for all $x \neq 0$

(ii) $f(x,y)f(y,x)=1,$ for all $x,y \neq 0$